We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.
Accepté le :
Première publication :
Publié le :
Keywords: Hyperbolic systems, operator semigroups, PDEs on networks, invariance properties, Saint-Venant system, second sound
@article{COCV_2021__27_1_A9_0,
author = {Kramar Fijav\v{z}, Marjeta and Mugnolo, Delio and Nicaise, Serge},
title = {Linear hyperbolic systems on networks: well-posedness and qualitative properties},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020091},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020091/}
}
TY - JOUR AU - Kramar Fijavž, Marjeta AU - Mugnolo, Delio AU - Nicaise, Serge TI - Linear hyperbolic systems on networks: well-posedness and qualitative properties JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020091/ DO - 10.1051/cocv/2020091 LA - en ID - COCV_2021__27_1_A9_0 ER -
%0 Journal Article %A Kramar Fijavž, Marjeta %A Mugnolo, Delio %A Nicaise, Serge %T Linear hyperbolic systems on networks: well-posedness and qualitative properties %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020091/ %R 10.1051/cocv/2020091 %G en %F COCV_2021__27_1_A9_0
Kramar Fijavž, Marjeta; Mugnolo, Delio; Nicaise, Serge. Linear hyperbolic systems on networks: well-posedness and qualitative properties. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 7. doi: 10.1051/cocv/2020091
[1] Regular solutions of transmission and interaction problems for wave equations. Math. Meth. Appl. Sci. 11 (1989) 665–685.
[2] Nonlinear Waves in Networks, Mathematical Research, vol. 80. Akademie Verlag, Berlin (1994).
[3] and , Study of the nodal feedback stabilization of a string–beams network. J. Appl. Math. Comput. 36 (2011) 441–458.
[4] and , Stabilization of elastic systems by collocated feedback. Vol. 2124 of Lecture Notes in Mathematics. Springer, Cham (2015).
[5] , , and , Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Comm. Pure Appl. Anal. 11 (2012) 785–807.
[6] and , Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Vol. 88 of Progress in Nonlinear Differential Equations. Birkhäuser, Basel (2016).
[7] , and , Positive Operator Semigroups. Vol. 257 of Operator Theory: Advances and Applications. Birkhäuser, Cham (2017).
[8] and , Multidimensional Hyperbolic Partial Differential Equations – First-order Systems and Applications. Clarendon Press, Oxford (2007).
[9] and , Spectral statistics for the Dirac operator on graphs, J. Phys. A 36 (2003) 2747–2769.
[10] and , The Selberg trace formula for Dirac operators. J. Math. Phys. 47 (2007) 112104.
[11] and , Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478.
[12] , , , and , Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111.
[13] , , , and , Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111.
[14] , Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer-Verlag, Berlin (2010).
[15] and , Parabolic systems with coupled boundary conditions. J. Differ. Equ. 247 (2009) 1229–1248.
[16] , Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc. 351 (1999) 4069–4088.
[17] , Nonclassical Sturm–Liouville problems and Schrödinger operators on radial trees. Electr. J. Differ. Equ. 71 (2000) 1–24.
[18] , Spectral theory for nonconservative transmission line networks. Netw. Heterog. Media 6 (2011) 257–277.
[19] , Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58 (1979) 249–273.
[20] and , Wave propagation, observation and control in 1-d flexible multi-structures. Vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006).
[21] , Semigroups for flows in infinite networks. Semigroup Forum 76 (2008) 341–356.
[22] , , and , The semigroup approach to transport processes in networks. Physica D 239 (2010) 1416–1421.
[23] and , The Berry–Keating operator on L2(ℝ$$) and on compact quantum graphs with general self-adjoint realizations. J. Phys. A 43 (2010) 095204.
[24] , Generator property and stability for generalized difference operators. J. Evol. Equ. 13 (2013) 311–334.
[25] and Exact and positive controllability of boundary control systems. Netw. Heterog. Media 12 (2017) 319–337.
[26] , , and , Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709–722.
[27] , , , and , Maximal controllability for boundary control problems. Appl. Math. Optim. 62 (2010) 205–227.
[28] and , One-Parameter Semigroups for Linear Evolution Equations. Vol. 194 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000).
[29] , Partial differential Equations – second edition. Vol. 19 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI (2010).
[30] , Momentum operators on graphs. In , , and , editors, Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday. Vol. 87 of Proc. Symp. Pure Math. Amer. Math. Soc., Providence, RI (2013) 105–118.
[31] , , , , , , , and , Observation of second sound in graphite at temperatures above 100 k. Science 364 (2019) 375–379.
[32] and , Quantum graphs with mixed dynamics: the transport/diffusion case. J. Phys. A 46 (2013) 235202.
[33] and , Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section. Appl. Num. Math. 79 (2014) 42–61.
[34] , and , C0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. J. Evol. Equ. 15 (2015) 493–502.
[35] and , Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces. Vol. 223 of Oper. Theory Adv. Appl. Birkhäuser, Basel (2012).
[36] , and , Momentum operators in two intervals: Spectra and phase transition. Compl. Anal. Oper. Theory 7 (2013) 1735–1773.
[37] , Difference operators as semigroup generators. Semigroup Forum 81 (2010) 461–482.
[38] , The flow approach for waves in networks. Oper. Matrices 6 (2012) 107–128.
[39] and , Kirchhoff’s rule for quantum wires. J. Phys. A 32 (1999) 595–630.
[40] and , Semigroups for dynamical processes on metric graphs. Phil. Trans. R. Soc. A 378 (2020) 20190619.
[41] , and , Hyperbolic systems with dynamic boundary conditions. Inpreparation (2020).
[42] and , Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249 (2005) 139–162.
[43] , Graph models of wave propagation in thin structures. Waves Random Media 12 (2002) 1–24.
[44] , and , Analytic solutions for stochastic hybrid models of gene regulatory networks. Preprint (2021). | arXiv
[45] , , and , Modeling, Analysis, and Control of dynamic Elastic Multi-Link Structures, Systems and Control: Foundations and Applications. Birkhäuser, Basel (1994).
[46] , Introduction to quadratic forms over fields. Vol. 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2005).
[47] and , Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13 (1960) 427–455.
[48] and , On the control of networks of vibrating strings and beams. In IEEE Conference on Decision and Control, IEEE, Providence, RI (1989) 2287–2290.
[49] and , A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15 (1967) 299–309.
[50] , Connecting of local operators and evolution equations on networks. In Potential Theory (Proc. Copenhagen 1979), edited by . Springer-Verlag, Berlin (1980) 230–243.
[51] and , A unified approach for the analysis of networks composed of transmission lines and lumped circuits. In Scientific computing in electrical engineering. Vol. 9 of Mathematics in industry. Springer-Verlag, Berlin (2006) 3–11.
[52] and , Asymptotic behavior of flows in networks. Forum Math. 19 (2007) 429–461.
[53] and , Symmetric bilinear forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Springer-Verlag, New York-Heidelberg (1973).
[54] , Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer-Verlag, Berlin (2014).
[55] , Polygonal Interface Problems. Vol. 39 of Methoden und Verfahren der mathematischen Physik. Peter Lang GmbH, Europäischer Verlag der Wissenschaften, Frankfurt/M (1993).
[56] , Control and stabilization of 2 × 2 hyperbolic systems on graphs. Math. Control Relat. Fields 7 (2017) 53–72.
[57] and , Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks Het. Media 3 (2007) 425–479.
[58] , Analysis of Heat Equations on domains. Vol. 30 of Lond. Math. Soc. Monograph Series. Princeton Univ. Press, Princeton, NJ (2005).
[59] and , Model of free electrons and the scattering problem. Theor. Math. Phys. 55 (1983) 485–492.
[60] , Semigroups of linear operators and applications to partial differential equations. Vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983).
[61] , On the spectrum of C0-semigroups. Trans. Am. Math. Soc. 284 (1984) 847–857.
[62] , Thermoelasticity with second sound – exponential stability in linear and nonlinear 1D. Math. Methods Appl. Sci. 25 (2002) 409–441.
[63] , Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Am. Math. Soc. 291 (1985) 167–187.
[64] , Hyperbolic partial differential equations and geometric optics. Vol. 133 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI (2012).
[65] , , and , Boundary systems and (skew-) self-adjoint operators on infinite metric graphs. Math. Nachr. 288 (2015) 1776–1785.
[66] , The Dirac Equation. Springer-Verlag, New York (1992).
[67] and , Dissipative extensions and port-hamiltonian operators onnetwork (2019).
[68] , Invariance of closed convex sets under semigroups of nonlinear operators in complex Hilbert spaces. SUT J. Math. 37 (2001) 91–104.
[69] , , and , Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16 (2010) 1077–1093.
Cité par Sources :
The work of M.K.F. was partially supported by the Slovenian Research Agency, Grant No. P1-0222, and the work of D.M. by the Deutsche Forschungsgemeinschaft (Grant 397230547). This article is based upon work from COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.
